| import OAI.Analysis.FiniteFactor.Sampling.Selection |
| import OAI.Analysis.FiniteFactor.CrossedProduct.PhaseDecay |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open Finset Filter |
| open scoped Topology |
| noncomputable section |
|
|
| def samplingQ (m : ℕ) : ℝ := (m : ℝ)^((7/10 : ℝ)*(m : ℝ)) |
| def samplingH (m : ℕ) : ℕ := m^300 |
| def samplingR (m K : ℕ) : ℕ := m*K*m^1000 |
|
|
| theorem sampling_integer_candidates (Q : ℝ) (hQ : 4 ≤ Q) : |
| Q/4 ≤ ((Finset.Icc ⌈Q/2⌉ ⌊Q⌋).card : ℝ) := by |
| have hf := Int.sub_one_lt_floor Q |
| have hc := Int.ceil_lt_add_one (Q/2) |
| have horder : ⌈Q/2⌉ ≤ ⌊Q⌋ := by |
| have : (⌈Q/2⌉ : ℝ) < (⌊Q⌋ : ℝ)+1 := by linarith |
| have ht : ⌈Q/2⌉ < ⌊Q⌋+1 := by exact_mod_cast this |
| omega |
| rw [Int.card_Icc] |
| have hcast : ((⌊Q⌋+1-⌈Q/2⌉).toNat : ℝ) = (⌊Q⌋ : ℝ)+1-(⌈Q/2⌉ : ℝ) := by |
| exact_mod_cast Int.toNat_of_nonneg (by omega : 0 ≤ ⌊Q⌋+1-⌈Q/2⌉) |
| rw [hcast] |
| linarith |
|
|
| theorem sampling_forbidden_budget (m K : ℕ) (hm : 4 ≤ m) (hK : K ≤ m) : |
| (((2*samplingH m+1)^2*(2*samplingR m K+1) : ℕ) : ℝ) ≤ (m : ℝ)^1605 := by |
| have hm1 : 1 ≤ m := by omega |
| have hh : 1 ≤ samplingH m := one_le_pow₀ hm1 |
| have hr : samplingR m K ≤ m^1002 := by |
| unfold samplingR |
| calc |
| m*K*m^1000 ≤ m*m*m^1000 := by gcongr |
| _ = m^1002 := by ring |
| have hp : 1 ≤ m^1002 := one_le_pow₀ hm1 |
| have hD : (2*samplingH m+1)^2*(2*samplingR m K+1) ≤ 27*m^1602 := by |
| calc |
| _ ≤ (3*samplingH m)^2*(3*m^1002) := by gcongr <;> omega |
| _ = _ := by unfold samplingH; ring |
| have h27 : 27 ≤ m^3 := (by norm_num : 27 ≤ (4 : ℕ)^3).trans (Nat.pow_le_pow_left hm 3) |
| have hhD : (2*samplingH m+1)^2*(2*samplingR m K+1) ≤ m^1605 := by |
| calc |
| _ ≤ 27*m^1602 := hD |
| _ ≤ m^3*m^1602 := Nat.mul_le_mul_right _ h27 |
| _ = _ := by ring |
| exact_mod_cast hhD |
|
|
| / |
| asymptotics, retaining Q, H, R and the exponent 0.69m exactly. -/ |
| theorem exists_source_sampling_powers (m K : ℕ) (hm : 200000 ≤ m) (hK : K ≤ m) : |
| ∃ N : Finset ℤ, |
| (∀ n ∈ N, samplingQ m/2 ≤ (n : ℝ) ∧ (n : ℝ) ≤ samplingQ m ∧ 0 < n) ∧ |
| (m : ℝ)^((69/100 : ℝ)*(m : ℝ)) ≤ (N.card : ℝ) ∧ |
| (N : Set ℤ).Pairwise (fun n n' => ¬SamplingConflict (samplingH m) (samplingR m K) n n') := by |
| have hmR : (200000 : ℝ) ≤ m := by exact_mod_cast hm |
| have hm0 : (0 : ℝ) < m := by linarith |
| have hm1 : (1 : ℝ) ≤ m := by linarith |
| have hQ : (m : ℝ)^1606 ≤ samplingQ m := by |
| unfold samplingQ |
| rw [← Real.rpow_natCast] |
| apply Real.rpow_le_rpow_of_exponent_le hm1 |
| norm_num |
| linarith |
| have hmQ : (m : ℝ) ≤ samplingQ m := by |
| calc |
| _ ≤ (m : ℝ)^1606 := le_self_pow₀ hm1 (by norm_num) |
| _ ≤ _ := hQ |
| have hQ4 : 4 ≤ samplingQ m := by linarith |
| have hR : (samplingR m K : ℝ) < samplingQ m/2 := by |
| have hr : samplingR m K ≤ m^1002 := by |
| unfold samplingR |
| calc |
| m*K*m^1000 ≤ m*m*m^1000 := by gcongr |
| _ = _ := by ring |
| have hsmall : (m : ℝ)^1002 * (m : ℝ) ≤ (m : ℝ)^1606 := by |
| rw [← pow_succ] |
| exact pow_le_pow_right₀ hm1 (by norm_num) |
| have hrR : (samplingR m K : ℝ) ≤ (m : ℝ)^1002 := by exact_mod_cast hr |
| have hpos : 0 < (m : ℝ)^1002 := by positivity |
| nlinarith |
| let S := Finset.Icc ⌈samplingQ m/2⌉ ⌊samplingQ m⌋ |
| have hS (n : ℤ) (hn : n ∈ S) : |
| samplingQ m/2 ≤ (n : ℝ) ∧ (n : ℝ) ≤ samplingQ m := by |
| have hh := mem_Icc.mp hn |
| constructor |
| · exact (Int.le_ceil _).trans (by exact_mod_cast hh.1) |
| · exact (show (n : ℝ) ≤ (⌊samplingQ m⌋ : ℝ) by exact_mod_cast hh.2).trans (Int.floor_le _) |
| obtain ⟨N,hNS,hpair,hcard⟩ := exists_sampling_separated (samplingH m) (samplingR m K) |
| (one_le_pow₀ (by omega)) S (fun n hn => by exact_mod_cast hR.trans_le (hS n hn).1) |
| refine ⟨N, fun n hn => ⟨(hS n (hNS hn)).1, (hS n (hNS hn)).2, ?_⟩, ?_, hpair⟩ |
| · have hh := (hS n (hNS hn)).1 |
| have : (0 : ℝ) < n := by linarith |
| exact_mod_cast this |
| · have hcount : samplingQ m/4 ≤ |
| (((2*samplingH m+1)^2*(2*samplingR m K+1) : ℕ) : ℝ)*(N.card : ℝ) := |
| (sampling_integer_candidates _ hQ4).trans (by exact_mod_cast hcard) |
| have hDb := sampling_forbidden_budget m K (by omega) hK |
| have hcount' : samplingQ m/4 ≤ (m : ℝ)^1605*(N.card : ℝ) := |
| hcount.trans (mul_le_mul_of_nonneg_right hDb (Nat.cast_nonneg _)) |
| have hexp : (m : ℝ)^1605*(m : ℝ)^((69/100 : ℝ)*(m : ℝ))*(m : ℝ) ≤ samplingQ m := by |
| unfold samplingQ |
| calc |
| _ = (m : ℝ)^(1605+(69/100 : ℝ)*(m : ℝ)+1) := by |
| rw [Real.rpow_add hm0, Real.rpow_one, Real.rpow_add hm0] |
| norm_num [Real.rpow_ofNat] |
| _ ≤ _ := by |
| apply Real.rpow_le_rpow_of_exponent_le hm1 |
| norm_num |
| linarith |
| have hprod : 0 < (m : ℝ)^1605 := by positivity |
| have htarget : 0 ≤ (m : ℝ)^((69/100 : ℝ)*(m : ℝ)) := by positivity |
| apply (mul_le_mul_iff_right₀ hprod).mp |
| nlinarith |
|
|
| / |
| def sourceSamplingPowers (s : ℕ) : Finset ℤ := |
| if h : 200000 ≤ sourceBlockSize s then |
| Classical.choose (exists_source_sampling_powers (sourceBlockSize s) |
| (sourceStageState s).K h (sourceStage s).m_ge_K) |
| else ∅ |
|
|
| theorem sourceSamplingPowers_spec (s : ℕ) (hs : 200000 ≤ sourceBlockSize s) : |
| (∀ n ∈ sourceSamplingPowers s, |
| samplingQ (sourceBlockSize s)/2 ≤ (n : ℝ) ∧ |
| (n : ℝ) ≤ samplingQ (sourceBlockSize s) ∧ 0 < n) ∧ |
| (sourceBlockSize s : ℝ)^((69/100 : ℝ)*(sourceBlockSize s : ℝ)) ≤ |
| ((sourceSamplingPowers s).card : ℝ) ∧ |
| ((sourceSamplingPowers s : Finset ℤ) : Set ℤ).Pairwise (fun n n' => |
| ¬SamplingConflict (samplingH (sourceBlockSize s)) |
| (samplingR (sourceBlockSize s) (sourceStageState s).K) n n') := by |
| unfold sourceSamplingPowers |
| rw [dite_eq_left hs] |
| exact Classical.choose_spec (exists_source_sampling_powers _ _ hs (sourceStage s).m_ge_K) |
|
|
| theorem sourceSamplingPowers_eventually : ∀ᶠ s in atTop, |
| (∀ n ∈ sourceSamplingPowers s, |
| samplingQ (sourceBlockSize s)/2 ≤ (n : ℝ) ∧ |
| (n : ℝ) ≤ samplingQ (sourceBlockSize s) ∧ 0 < n) ∧ |
| (sourceBlockSize s : ℝ)^((69/100 : ℝ)*(sourceBlockSize s : ℝ)) ≤ |
| ((sourceSamplingPowers s).card : ℝ) ∧ |
| ((sourceSamplingPowers s : Finset ℤ) : Set ℤ).Pairwise (fun n n' => |
| ¬SamplingConflict (samplingH (sourceBlockSize s)) |
| (samplingR (sourceBlockSize s) (sourceStageState s).K) n n') := by |
| filter_upwards [sourceBlockSize_tendsto.eventually (eventually_ge_atTop 200000)] with s hs |
| exact sourceSamplingPowers_spec s hs |
|
|
| end |
| end FiniteFactor |
|
|
| end OAI |
|
|
